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Journal of Marine Research 45293-3181987

Observations on the vertical structure of tidal and inertialcurrents in the central North Sea

by L R M Maas1 and J J M van Harenl

ABSTRACTTidal and inertial current ellipses measured at several locations and depths in the central

North Sea during a number of monthly periods in 1980 1981 and 1982 are decomposed intocounterrotating circular components to which Ekman dynamics are applied to determineEkman layer depths and vertical phase differences from which are inferred overall values of theeddy viscosity and drag coefficient Stratification effects produce an additional vertical phaseshift of the anticyclonic rotary component indicative of an inverse proportionality of the eddyviscosity to the vertical density gradient From the time variations of the Ekman layer depths ofthe semidiurnal tidal components as well as from the vertical structure of the inertial currentcomponent we infer variations in the relative vorticity of the low-frequency flow

1 IntroductionThe effects of bottom friction on a steady current in a rotating frame of reference

(governed by Ekman dynamics eg Pedlosky (1979)) and on an oscillating current in aresting frame (eg Lamb 1975) bear similar features Both show an amplitudedecrease toward the bottom accompanied by an anticyclonic veering with depth in theformer case of a steady current and a phase advance toward the bottom in the lattersituation of an oscillatory current (in the Northern hemisphere) Oscillatory currents(frequency 0) in a rotating frame should combine both aspects

The nonviscous effect of rotation on an oscillating current is to yield a secondorthogonal velocity component producing an ellipsoidal motion in a horizontal planeexemplified by plane Sverdrup waves The amplitude and phase of this extra velocitycomponent introduce two more degrees of freedom so that the ellipsoidal currentmotion is entirely described by four parameters

U maximum current velocity or semi-major axise eccentricity ie the ratio of semi-minor (V) to semi-major axis negative values

indicating that the ellipse is traversed in an anticyclonic senseif inclination or angle between east (x) direction and semi-major axiscp phase angle ie the time of maximum velocity with respect to a chosen origin of

time

I Netherlands Institute for Sea Research PO Box 59 1790 AB Texel The Netherlands

293

294 Journal of Marine Research [452

crt-t=O (f) +

Figure 1 Decomposition of a tidal current ellipse (frequency IT) specified by the four ellipseparameters magnitude of semi-major axis V eccentricity e - VIV (V semi-minor axis)inclination 11 and phase angle fjJ into two counterrotating currents of constant magnitudes Wand directions (J bullbull

Following Prandle (1982) to understand the vertical structure of these fourparameters we decompose the ellipsoidal motion into two counterrotating circularvelocity components with fixed amplitudes (the radii W) and phases (0) Figure 1 interms of which the ellipse parameters read

u= W+ + W_

e = (W+ - W_)(W+ + W_)

1 = (0_ + 0+)2

cent = (8_ - 0+)2

Sverdrup (1927) pointed out that in terms of these rotary current parameters thegoverning (linear) equations are solved in a straightforward way It is conceptuallyadvantageous however to stress that this solution procedure (Section 2) consists oftwo steps Firstly the circular velocity components which themselves are given on auniformly rotatingf-plane are transformed to two co-rotating frames having differentangular velocities v2 = (f plusmn u)2 where f denotes the local inertial frequencySecondly since in their respective co-rotating frames the velocity vectors reduce tosteady currents merely giving the amplitude and angle with respect to an originallychosen direction Ekman dynamics can be applied directly This explicit separationallows us to obtain a qualitative mental picture of the causal relations underlying thevertical structure of the ellipse parameters

Thus for super inertial frequencies (u gt f) the net rotation sense of the frameco-rotating with the anticyclonic current component is negative (clockwise) Hencedynamics apply as if we are on the Southern hemisphere and therefore the anticyclonicvelocity vector will rotate clockwise toward the bottom For semidiurnal frequencies atmoderate latitudes (u lt f) a separation of Ekman layer scales 0 = y2K If plusmn ui(Soulsby 1983) is predicted 0+ laquo 0_ Here K denotes the turbulent eddy viscosityNear the bottom therefore the cyclonic current component will be less reduced than

1987] Maas amp van HarenVerticalcurrent structure 295

its anticyclonic counterpart This implies that a near-surface rectilinear currentacquires an elliptical shape traversed in a cyclonic sense closer to the bottom

For subinertial frequencies (0- ltf) the net rotation sense of both co-rotating framesis positive (anticlockwise) hence currents will rotate anticlockwise toward the bottomfor both components although (for the same reason as above) there may be a distinctdifference in scales between them

The anticyclonic component of the inertial oscillations (0- = f) is by its nature givenin a nonrotating frame Inspection of the governing equations predicts some sort of az2-profile due to bottom friction (where z is the vertical distance above the sea bed) andthe absence of any turning with depth

2 TheoryThe equations of motion describing the dynamics of tides in a homogeneous sea in

the presence of turbulent friction read (nondimensionally) (Sverdrup 1927)

au at E a2u(Ia)--v+-=--

at ax 2 az2

av at E a2v(lb)-+u+-=--

at ay 2 az2

~ + 7 bull (f U dZ) = 0 (Ie)

auz=o (Id)-~smiddot U at

az

au(Ie)-=0 at z = 1

az

The vertical coordinate z measured upward from the top of the bottom boundarylayer is scaled with the local depth H Horizontal coordinates x and y correspondingto east and north directions are scaled with the barotropic Rossby deformation radiusR = Jiiif where g denotes the acceleration of gravity Time t is scaled withf-IHorizontal velocities u and v corresponding to currents in x and y directions are scaledwith a typical velocity magnitude [u] The surface elevation t is nondimensionalizedwith [u] Jiii x H The only two remaining nondimensional parameters are thefamiliar Ekman number E = 2KfH2 and the stress parameter s = rHK with r abottom friction velocity (r = O(I0-4m S-I) Csanady 1982) related to the dragcoefficient Cd (Cd = 2 - 4 X 10-3 Bowden 1983) The stress parameter variesbetween no-stress (s - 0) and no-slip ie infinite stress (s - (0) when the flow stickscompletely to the bottom The bottom boundary condition (Id) is derived at the top of


the bottom boundary layer (z = 0) circumventing a detailed description of this shallowboundary layer of approximately 1 m depth (Bowden 1983) by equating interiorstress K ouoz to bottom stress Cdl ul u or more preferably its linearized counterpartro The connection between r and Cd can be established by an energy criterionrequiring the dissipation averaged over a tidal cycle to be equal (Lorentz 1926) andgives r = 831r x CdU(O) where U(O) denotes the tidal current amplitude at thebottom The correct spatial and temporal dependence of the eddy viscosity constantK is an intensively studied subject in the tidal context (Tee 1979 Prandle 1982 Fangand Ichiye 1983) Since in this paper the diffusion process is studied over large spatialareas from observations taken over monthly periods in different years the parameter-ization should lose its sensitivity to the detailed structure of the flow field and aconstant eddy viscosity is therefore adopted

By assuming u = RI(u exp (-i(Jtraquo where (J = (Jf u = U(x) bull exp (ictgtu(xraquo andsimilarly for v and S we may obtain two independent second order differentialequations by forming the complex velocities

u + iiiw ~ -- = W exp (i8 )- 2 - -

u - iiiw = -- = W exp (- i8 )+ 2 + +

(2)

related to the ellipse parameters discussed in the introduction These velocities w _ andW + act as the amplitudes of the circular rotary current components which traverse theunit circle in an anticyclonic and cyclonic sense respectively as can be seen when wecombine the horizontal velocities u and v directly into a complex velocity w

w = u + iv = U cos (rgtu- (Jt) + iV cos (rgtv- (Jt)

= w_ exp (-i(Jt) + w exp (iut) (3)

where the asterisk ( ) denotes a complex conjugate In terms of these velocities theequations of motion (la b) become omitting the common factor exp (plusmni(Jt)

IE d2W~1(1 - u)w + - jr= ---~ 2 ~ 2 dz2

with boundary conditions

dw+-- =swdz plusmn

dwplusmn = 0dz

-1(1 ) I r E d2w++(JW +-j~=---

+ 2 2 dz2

at z = 0

at z = I

(4)

(5)

1987]

where

Maas amp van Haren Vertical current structure

a a a av = - + i-and v = - - i - ax ay ax ay

297

These equations are equivalent to the (complex) equation describing the dynamics of asteady current on a plane uniformly rotating at a rate v2 (Ekman dynamics Pedlosky1979)

with

(6)

dw-=swdz

dw-=0dz

at z = 0

at z ~ 1

(7a)

(7b)

when replacing the Ekman number E = 2KvH2 by either E(I - u) or -E(I + u)and the pressure gradient vp by either 2vt(I - u) or _12 vr(I + u)respectively The unconventional choice of the inertial frequency v has been adoptedto allow for a direct application to frames rotating with rates (I plusmn u)2

The mathematical equivalence of the dynamics governing the rotary components (4)to those that govern a steady current (6) implies the underlying transformation toco-rotating co-ordinate frames Borrowing the Ekman solutions to (6) (with a modifiedbottom stress condition (7a))

where

w = ivp

we obtain the solutions of (4-5) as

cosh az )cosh a + (as) sinh a

a = (I + i)E~2

(8)

(9)

_ ( cosh aplusmnz )w = w 1 - ----------plusmn plusmn cosh aplusmn + (aplusmns) sinh aplusmn

where

(10)

(II)

298

and

Journal of Marine Research [452

a_ = [(1 - i) (Hjo_)(1 + i) (Hjo_)

a+ = [(1 - i) (Hjo+)(1 + i) (Hjo+)

poundI gt 1

1gt poundIf

poundIfgt -1

-1 gt poundIf (12)

in terms of Ekman layer depths (Souls by 1983)

oplusmn = ~2Kj (If plusmn poundII) (13)

Solutions (10) finally are used in the comparison to the observed profiles in Section 4Their qualitative features are discussed in the introduction

3 The data acquisitionDuring the spring summer and autumn of 1980 1981 and 1982 a collaborative

study was performed in the stratified central North Sea by the Royal NetherlandsMeteorological Institute (KNMI) the Institute of Meteorology and OceanographyUtrecht (IMOU) and the Netherlands Institute for Sea Research (NIOZ) InFigure 2 local bathymetry and current meter mooring positions are shown Table 1gives the precise operational periods and depths The current meter records weresupplemented with extensive hydrographic surveys on both large (0(100 kmraquo andsmall (0(20 kmraquo horizontal scales A limited number of thermistor chains andpressure gauges completed the data equipment Onset evolution and decay of thestratification (van Aken 1984 1986) and frontal dynamics (van Aken et al 1987)have been discussed elsewhere

The current measurements presented here were performed in both stratified andwell-mixed periods Local water depths H are between 45 and 50 m the measurementsite being a relatively flat area Each of the time series of current measurements hasbeen harmonically analyzed using a small number of well separated tidal frequencies(01 Kio N2 M2 S2 M4 M6) and the inertial frequency f(Dronkers 1964) In view ofthe presence of stratification some internal tide contamination may be anticipatedSince the observation site is far away from any topographic features we assume thatfree internal tides will not be phaselocked to the surface tide and hence in view of thelarge periods used will be filtered out due to their intermittent character (Wunsch1975)

Since small depth differences at different moorings result in local differences in tidalamplitudes the current measurements were normalized by their weighted depth-averaged values (the weighting factor of a current meter being proportional to thedepth increment which it represents divided by water depth) while phases were takenwith respect to those from the current meter near the surface

N56deg

51degI I I I I I I

0deg 1deg 2deg 3deg 4deg 5deg 6deg E

UJ wCo rC) It0 degC) C)

AD 8+1- 54deg46N

1980 10km

0+ c+J 54deg41N

UJ UJ wN 0 CoCI C) (t)

deg 0 0It It ltt

1981 F-jamp~ 54deg35 N

EDI

16km

1982H~

54deg30 N

GDK 54027 N

Figure 2 (a) Map of part of the North Sea with the positions of mooring networks in 1980 (A)and 1981 1982 (8) with isobaths in m Wind observations are performed at platform labelledK13 (b) Mooring configurations in 1980 1981 and 1982 Current meters are denoted by +thermistor chains by O


Table 1 Positions current meter depths local sea depth and operational periods of the currentmeasurements in 1980-1982

Water PeriodStation N E Depths (m) depth (m) (year-day) Year

A 54deg46 3deg38 16 46 240-289 198041 240-275

B 54deg46 3deg47 14 46 240-3052841 240-275

C 54deg41 3deg47 13 46 240-27541 240-273

D 54deg41 3deg38 27 46 240-27341 240-274

E 54deg30 4deg30 1319313845 50 133-155 1981F 54deg35 4deg31 132945 50 133-155G 54deg27 4deg22 132945 50 133-155H 54deg28 4deg38 122842 47 133-155I 54deg30 4deg30 12243037 49 215-252 1982

18 215-23944 215-240

J 54deg35 4deg30 1243 48 215-252K 54deg27 4deg22 122744 49 215-252L 54deg27 4deg38 122742 47 215-252

The bottom current meter at G 1981 was eliminated due to compass failure

4 Observational versus tbeoretical current profiles

a The superinertial tidal frequency bandBottom friction imposes a number of typical features on the vertical structure of the

tidal currents as demonstrated in Figure 3 for the observed M2 component at positionI (Table I) From the surface downward we observe

a slight increase in maximum current amplitude followed by a sharp decreasetoward the bottom

a clockwise turning of the major axis

an increase in the eccentricity of the ellipse

a phase advance indicating that maximum current speeds are reached earliernear the bottom

These features summarized here for a single span of time at a specific geographicposition are directly visible for all observational periods and positions listed in Table 1from graphs showing the ellipse parameters versus depth which are plotted inFigure 4 for the M2 frequency Transforming these ellipse parameters to amplitudesand phases of the rotary circular components confirms experimentally the expected

1987] Maas amp van Haren Vertical current structure 301

12 m

18 m

24 m

o cmS

--

N

n

30 m

37 m

44 m

-

Figure 3 Observed M2-current ellipses at position 1 (Fig 2) in 1982 for the period listed inTable 1 as a function of depth The angle which each straight line makes with true North givesthe phase angle rfgt with respect to the beginning of the year of observation


5

-~ ~ UO - e I -Or If Or f

Figure 4 Observations (denoted by dots) of ellipse parameters for the M2-frequency versusnormalized depth Maximum current (U) is divided by iJ the weighted depth averagedcurrent amplitude Phases are given with respect to their surface values Solid curves are bestfit theoretical curves as derived in Section 2 Error bars are indicated at the top

separation of vertical friction scales (Fig 5) Phases turn in opposite directions towardthe bottom in agreement with predictions from Ekman dynamics for frames rotatingin opposite directions

The theoretical curves (10) are given by the solid lines in Figures 4 and 5 Themagnitude of the constants E and s required in (10) have been determinedexperimentally by choosing a best fit by face value of the theoretical curves to theobservations Note that this best fit determination has to be done simultaneously foreach set of four graphs in Figures 4 and 5 The experimental values are E = (10 plusmn

01) x 10-2 and s = 10 plusmn 1 from which we deduce for a depth H = 48 m anexperimental value of K = (14 plusmn 01) x 1O-3m2 S-1 andr = (29 plusmn 02) x 1O-4ms-lbullFrom the theoretical profile (10) we find a time-averaged bottom velocity magnitudeU(O) of 055 times a vertically averaged tidal velocity amplitude of 25 em S-I Thisresults in Cd = (25 plusmn 02) x 10-3bull This value of the drag coefficient agrees with atypical value (Bowden 1983) although the friction velocity r is a little less thansuggested by Csanady (1982) The value of the eddy viscosity K falls below theoreticalestimates of K = kEu20f (where the friction velocity u = cj2 U(O)) appropriate for

8 Or -OrIIw_ 2 -60-o

o

05

ZlH

Figure 5 Observations (denoted by dots) of amplitudes W and phase angles 8 of the M2

frequency rotary current components as a function of normalized depth Solid curves are bestfit theoretical curves Error bars are indicated at the top_


0

o w_ 2 Iilt if

J)

bull to __

9_ liltf

Figure 6 As Figure 5 but for the S2 frequency

situations where kEulf lt H (Csanady 1976) There is some uncertainty about thevalue of the constant kE which varies between 01 (Csanady 1976) and the vonKarman constant 04 (Wimbush and Munk 1970) With kE ~ 01 we obtain anestimated K = 16 X 10-3 m2 S-I An inferred boundary layer thickness (a weightedcombination of the two Ekman depths oplusmn) is below 20 m and in this sense agrees withvalues given by a map of this parameter by Soulsby (1983) at our mooring location

Finally the value of the nondimensional stress parameter s raquo 1 indicates that theno-slip Ekman boundary condition is more appropriate than a no-stress condition

The most important deviations between observed and theoretical ellipse parametervalues occur at the bottom current meter and may be due to either an incorrect depthdetermination (probably less than 05 m) to which the sheared current profile isparticularly sensitive near the bottom or a local breakdown of the constant Kassumption which according to Prandle (1982) should be replaced by an eddyviscosity increasing from the bottom upward recognizing the increase in mixing-length

Errors in the observed ellipse parameters denoted by error bars were estimatedusing Tees (1982) method with an overall root-mean-square value of the residualcurrents (defined as original minus harmonic time series) of u = 5 cm S-I This valueimplies an overall error in the estimated harmonic amplitude of the Cartesian velocitycomponents of0 = ul MI2 = 025 cm S-I where M denotes half the number of hourlydata points (typically M = 400 Table I) The errors in the ellipse parameters areinversely proportional to the amplitudes of the appropriate counterrotating currentcomponents

The behavior of the vertical profiles of other semidiurnal frequencies (S2 and N2) issimilar to that of the M2 components (Figs 6 and 7) Their smaller energetic content(S2 = 0(7 cm S-I) N2 = 0(4 cm S-Iraquo is reflected in a larger scatter around the meanprofiles This scatter tends to be larger for the anticyclonic component despite the factthat the energy is fairly well distributed over the two rotary components (for S2 W+ =39 cm S-I W_ = 32 cm s- while for N2 W+ = 23 cm S-I W_ = 20 cm S-I) This isprobably due to a larger influence of stratification effects on the anticycloniccomponent as will be examined in the next subsection for M2bull The theoretical curves

304

ZHbull0

-

Journal of Marine Research

bullbull II bullbull

[452

oo W 2 o

w_ 2 -04 8_ 04

Figure7 As Figure5 but for the N2frequency

used in these figures are determined by using the best fit parameter values obtainedfrom the M2-fit and confirm their application to these frequencies too The differencecaused by a changing frequency is as expected small and curves for M2S2and N2arenearly indistinguishable implying that friction operates in a similar way in thissemidiurnal band

Theoretically there is a difference with the profiles calculated for the M4 frequencyObserved amplitudes however are small (W+ = 054 cm S-I W_ = 054 cm S-I)

(Fig 8) and one must be cautious in drawing conclusions The calculated error barsexceed any variance suggested by the observations by a factor two This implies thatthe noise level is smaller in this frequency range and therefore the noise is probably notwhite as the theoretical estimate (Tee 1982) assumes

Dependence of eddy viscosity on stratification For the M2 frequency the highsignal-to-noise ratio makes it worthwhile examining the deviations from the theoreticalcurves (Figs 4 and 5) which we believe to be related to stratification effects

In August-September 1982 this could be checked since a frontal passage wasrecorded separating a period (day 215-230) with a well-defined thermocline (tempera-ture jump 9degC) at about 25 m depth from a weakly stratified period (day 237-252)where the thermocline (temperature difference 3degC) was at some 40 m depth Thelow-passed temperature field at the central mooring I (Fig 9a van Aken et al bull 1987)

ZlH0

0

bull

e

oo W 2 o w_ 2 -04 8 04 -04 8_ 04

Figure 8 As Figure5 but for the M4 frequency


245 250

TIME (DAYS)

230 235

10

125120

11

12

215

]0

50

II

II I NO OBSERVATIONSIibullbull IQ

I10 I

1--------

20

N

nt------t5 d y n c m2

215 235YEAR-DAY 255

Figure 9 (a) The low-passed temperature field at mooring I (1982) Isotherms are given in 0c(From Van Aken et af 1987) (b) Observed windstress (-Cdlualua) using the oceanographicconvention at platform K13 (Fig 2)


8_ rid

[452

I

bull tI

II lll171D1

if2 -iiiw_

II 11111 7lll

W 2

306

Figure 10 Observed counterrotating Mrtidal velocity components for the stratified (day215-230 +) and unstratified (day 237-252 6) periods (Fig 9) The solid line gives theanalytical result of a three-layer model with a small eddy viscosity value K = 6 x 10-4 m2 S-I

in the middle layer These curves may be compared with the original curves from Figure 5dashed here The hatched area represents the modelled thermocline

is interpreted from hydrographic surveys as a gradual deepening of the surface mixedlayer (day 215-230) due to wind-mixing (Fig 9b) followed by a subsequent period(day 230-238) of combined wind-mixing and frontal advection In Figure 10 wecontrast results from a harmonic analysis of the two periods Note that the observationsfor the unstratified period (day 237-252) are biased by the absence of the bottomcurrent meter information from day 240 onward Free internal tide motions can beexcluded as being the source of the observed deviations since horizontal coherenceanalysis of the residual signals (defined as original minus tidal harmonic time series)showed very small phase differences between the different moorings (20deg) which isat variance with a theoretically expected value of approximately 180deg for the givenstratification and mooring distances (Schott 1977)

Figure 10 shows that deviations from the theoretical curves in Figure 5 (dashedhere) occur in the anticyclonic component only The inference is that the cyclonicEkman layer depth is so small that the bottom Ekman layer is well separated from theEkman layer at the pycnocline In contrast the anticyclonic Ekman layers frombottom interface and (possibly) surface apparently interact

A density jump such as that occurring during the stratified period is believed tosuppress turbulent motions within the pycnocline inhibiting momentum transfer andresulting in a slab layer motion Following Fjeldstad (1963) it is appropriate to modelthe eddy viscosity in inverse proportion to the density gradient To idealize thegeometry we consider a three-layer viscous model in which the eddy viscosity in themiddle layer is an order of magnitude less than that of the homogeneous bottom andsurface layers (Sverdrup 1927) For simplicity we assume the eddy viscosity in thelatter layers to be equal The equations of motion in each of the layers are equivalent to(4) subsequently applied to frames rotating with the anticyclonic and cycloniccomponents implying that Ej_ ~ Ej(u - 1) 5 x Ej Ej+ = Ej(u + 1) 05 x Ej(j = 1 2 3) for u 12 Requiring velocity and stress continuity at the bottom and top

1987]

ZlHLO

Maas amp van Haren Vertical current structure 307

O

oo It 2 It_ 2 -0

Omiddot e- d

Figure 11 As Figure 5 but for the 01 frequency

of the interior layer one may find the analytical solution of the current profiles drawnin Figure 10 For convenience we adopted a no-slip bottom and stress-free surfaceboundary condition The best fit values of the Ekman numbers determined again byface value are E = E3 = 15 X 10-2 and E2 = 5 X 10-4 setting the middle layerthickness equal to 018 close to the observed pycnocline thickness

The resulting calculated phase jump of approximately 300 for 8_ at the bottom of theinterior layer is in accordance with the observed phase jump thus confirming thelikeliness of a reduced eddy viscosity value in stratified circumstances as has beenverified from other data sets (eg Gargett and Holloway 1984)

The amplitude profiles generally correspond well to the observations but fail todescribe the amplitude decrease in the anticyclonic component toward the surfaceThis is probably due to a failure in the applied zero surface stress condition since ingeneral it reads T = K iJu iJz = Cd IUo - u I(Uo - u) where Uo is the velocity of the airwhich under calm weather conditions (uo laquo u) reduces to K iJuiJz = -Cdlulu Thismay again be linearized as for the bottom stress condition to iJujiJz = -)11 with l anondimensional surface stress parameter which we took to be zero up to here Notethat since we are looking at the current profile u of one specific tidal component thewind speed Uo also refers to the contribution of the wind in this frequency range onlygiving some wider range of applicability to the calm weather condition Applying thelatter surface stress condition qualitative agreement in amplitude decrease is obtainedfor y = 1 This both shows that the wind has little persistent influence on the M2-tideand that surface friction may not always be negligible

The observed values of the surface and bottom stress parameters allow thedetermination of typical stress magnitudes using a velocity magnitude of O( 10 cm s-1)Tsurface = 002 dyn cm-2 and Tbollom = 02 dyn cm-2 (compare with a typical surface windstress T = 2 dyn cm-2 based on Uo = 10 m s-)

b The subinertial tidal frequency bandObserved and theoretical profiles for the diurnal constituents 0 and K] are given in

Figures 11 and 12


ZlH10

oo

w 2

if

8 till

Figure 12 As Figure 5 but for the K1 frequency

In both co-rotating frames the theoretically expected cyclonic rotation of thecurrent vectors toward the bottom is more or less confirmed at both subinertialfrequencies although the observations can hardly be used to support the predictedratio offriction depths 0+0_ = (f + u)(f - u) = 04 Both diurnal components arerotating predominantly cyclonic with time for 01gt W+ = 14 cm S-1 and W_ =

05 em S-I while for KIgt W+ = 12 cm S-1 and W_ ~ 06 cm S-I hence the large scatterin the anticyclonic components

Note that the theoretical curves have again been calculated using the parametervalues E = 001 and s = 10 obtained from the M2 fit which seem to be fairly suitable

c Inertial oscillationsThe observed tidal current components do not show the influence of stratification

except for the effects on K treated in Section 4a The observed inertial oscillations arean exception at this point and seem to be purely baroclinic in nature In Figure 13 thisdifference is demonstrated for observations in May 1981 with a mean thermoclinedepth at 15 m For mooring E the current ellipses are drawn resulting from harmonicanalyses for Ofand S2 (with different scaling) respectively The 1800 phaseshift infbetween the current ellipse near the surface and the lower four current ellipses clearlyindicates the presence of first mode baroclinic inertial motion The inertial currentellipses are almost circular rotating in an anticyclonic sense in time This reflects theresonant response with which the anticyclonic inertial oscillations react to a broadfrequency band wind forcing (Pollard and Millard 1970) This inertial oscillation wasalso present at the other three moorings in 1981 showing marginal phase shiftsbetween current meters at comparable depths A harmonic analysis was performed forthe current meter records in 1982 for both the stratified and unstratified periodFigure 14 demonstrates the dramatic decrease in amplitude of the internal inertialoscillation in the second unstratified period despite the fact that the wind forcing isnot significantly less than over the stratified period (Fig 9b)

The 1982 observations (Fig 14) demonstrate more clearly than those of 1981(Fig 13) the influence of bottom friction which causes a strong reduction in amplitude

1987] Moos amp van Haren Vertical current structure 309

N

2 cms n 2~gt----lt

13m -Q -CJ19 m Q G)31 m Q Q38m Q Q45 m Q Q

0 f

7~

-

(14)

Figure 13 Observed current elIipses for 01gt f and S2 at position E for the period in 1981 listed inTable 1 as a function of depth The phase angles cP with respect to true North are given by astraight line Note the differences in scales The depth of the thermocline is indicated by adashed line

toward the bottom together with a phase lead indicative of downward energypropagation (Gill 1984)

In Figure 15 we have plotted the observed (dominant) anticyclonic inertialamplitudes and phases for the 1982 data as a function of depth The net rotation rate ofthe frame v2 = (I - q)2 approaches zero In this limit the velocity profile w_ from(10) approaches with v = 1 - q

Vs iV hm w_ = - z(z - 2) 1 + - (Z2 - 2z - 4) + O(V2)-0 2pound 6pound

for the more transparent no-slip bottom boundary condition (s -+ 00) Thus abarotropic inertial wave acquires a z2-structure Clearly the observations imply thatthe inertial wave mode is of a baroclinic rather than barotropic nature Hence if thedensity structure is modelled with a profile linearly increasing downward the pressuregradients may also have a vertical structure imposed by anyone of the internal modes

310 Journal of Marine Research

N

12m ~ 0 0 018m C)

G3crrvS-

24m -Q--Q -------27m

30m

37m Q

44m C Q G

J K L

12m ~ lt2gt ()

Jcmls---24m

27m

30m

37m

44m () -I J K L

[452

Figure 14 Observed current ellipses as a function of depth for inertial frequency motion duringthe stratified (a) and unstratified (b) periods in 1982 Phase angles 1gt with respect to trueNorth are given by the straight lines The depth of the thermocline in the stratified period isindicated by a dashed line

1987]

ZlH10

5

00


L

20 -1 Bd 90middot

311

1 Bo

Figure IS As Figure 5 for the anticyclonic inertial current components in 1982 Theoreticalbarotropic (dashed) and first mode baroclinic (dashed-dotted) profiles correspond to (14) and(15) Section 4c respectively The solid curve represents a first mode baroclinic profileobtained with an apparent Ekman number E = 2KvH2 = 01 and stress parameter s = 10

vfmiddot cos n1lz (Wunsch 1975) In particular a first mode internal wave gives again inthe no-slip limit for vanishing frame rotation rates

vf 11 iv hm w_ = -2- sm2 - Z + -z(z - 2) + 0(V2)v-o 11 E 2 E (15)

Neither of these profiles seem to offer a sufficient description (Fig 15) Note that theyespecially predict a zero phaseshift in the vertical The reason for this bad fit is thatmodification of a baroclinic current is very sensitive to deviations of the frequency fromfbecause the apparent Ekman number E = E flv multiplies the large parameter flvwith the Ekman number E = 2KlfH2

which itself is a small number of 0(10-2)

(Section 4a) Therefore slight deviations of v away from zero of 0(10-1 x f) maygive rise to small values of the apparent Ekman number E instead of leading to aninfinite E which formally applies in the limit u -- f (v -- 0) considered above WithE = 0(10-1

) a qualitative agreement is indeed obtained between the observationsand a frictionally modified first mode baroclinic wave in a linearly stratified sea (thesolid curve in Fig 15) pointing at the presence of low-frequency relative vorticity Thecrudeness of the adopted model of the density profile demonstrates that no more thanqualitative agreement is pursued here However the model confirms the observedrotation of the anticyclonic current vector of about 2200 toward the bottom which adds


Table 2 Amplitudes and phases of surface current meters for the inertial frequency atmoorings in 1982

Position (1982) W+ (cms) W_ (cms) 0+ O_

J 05 18 27deg -18degJ 06 23 71deg -31degK 02 18 _11deg -43degL 02 25 15deg _7deg

the frictional effect of some 40deg to the 180deg phase jump of a first mode baroclinicwave

The relatively small phase differences at comparable depths between the fourmoorings (Table 2) (for the anticyclonic component less than 40deg) indicate that there isa strong coherence between the time series and that the wavelength of the inertial wavegreatly exceeds the mooring separation distance of 0(10 km) This is in accordancewith results found by Schott (1971) who estimated a wavelength of 0(50 km) for anarea just north of the Dogger Bank The propagation direction suggested by the phasevalues in Table 2 is east-south-east and points to the Dogger Bank as source area as inSchotts study

That inertial oscillations can be captured with a sharp spectral filter like harmonicanalysis indicates the persistence of these motions although the values obtained mustrepresent an average This is illustrated in Figure 16 where we show the amplitude ofthe dominating anticyclonic motion after applying a bandpass filter with half widthpoints at 10 x 10-4 S-I and 12 x 10-4 s- to the spectrum of the original data setMaximum inertial currents reach up to 8 cms -I Notice the increase of the inertialwave amplitude at the deeper current meters in the frontal zone between day 230-235(Fig 9) Kunze and Sanford (1984) attribute an increase in frontal inertial energy to atrapping of inertial waves due to a local decrease in the effective Coriolis frequency(Mooers 1975) fo = f + V2[(avojax) - (auojay)] associated with the sheared frontaljet (uo vo) Also note the slow amplitude modulations in Figure 16 with a typicalperiod of 5 days which may not be an artifact due to the filtering procedure but can beattributed to the beat (0- - f) of the actual near-inertial frequency and the exactinertial frequencyf(GiII 1984)

Inertial motions are either directly forced by wind stress (Pollard and Millard1970) or appear as the transients in a geostrophic adjustment process which due totheir low phase speeds remain near the source (Gill 1984)

The specific frictionally modified first mode appearance of the inertial motions isattributed to the slab-like motion of the surface layer which is the initial response towind forcing (Millot and Crepon 1981) The relative forcing of the internal modes issubsequently determined by the projection of this step-function forcing on the(normal) modes yielding a strongest forcing for the first mode

1987]

~ 0Ev


YEAR-DAY

12m

313

18m

10

o

10

o

10

o

24m

JOm

37m

44m

Figure 16 Slow time evolution of band-passed anticyclonic inertial current amplitude W_ as afunction of depth for position I in 1982

5 Slow time e~olution offrictional depths hplusmn and phase differences L8plusmnA variation in the effective Coriolis frequency at a slow time scale should

immediately be mirrored in variations in Ekman depths and total amount of phaseshifts between surface and bottom currents for the semidiurnal rotary tidal currentcomponents For this reason we determine from the observations the slow timeevolution of Ekman depths bplusmn and phase shifts L8 plusmn (the hat indicating the experimentaldetermination) which are defined as the depth where the velocity obtains an(arbitrary) 08 value of its surface magnitude and as the difference between phaseangles at the surface (Os) and bottom current meters (Ob) respectively The depth b isset zero if the 08 value falls below the depth of the lowest current meter Now since theEkman depth 0 ~ 2Kjv it formally approaches infinity when v - 0 clearly a featurewhich cannot be simulated using the experimental definition We therefore calculated


10

5

0-100 -50 o

F50 100

-40middot-100 -50 o

F50 100

Figure 17 Profiles of B(a) and 68(b) as a function of frequency v scaled with the Ekmannumber E (F = viE) for the theoretical Ekman solution (8)

b from the theoretical Ekman profile (8) with a no-slip bottom boundary condition asa function of frame rotation rate P scaled by the Ekman number E (Fig 17a) Alsosince the bottom current meter in the experimental setting is at an average height of01 x H we calculated from the Ekman solution60 = 8(1)-0(01) (Fig 17b) Notethat there exists some ambiguity in 60 values since similar values can be obtained atdifferent frequency ranges P

The observed bplusmn and 60plusmn are obtained by performing a harmonic analysis oversubsequent two-day periods In view of the shortness of these periods we cannotdistinguish between different semidiurnal frequencies (M2 S2 N2) which are there-fore lumped together As shown in Section 4a this has no consequence for the verticalprofiles since it is the frequency (J which determines the behavior of bplusmn and 60plusmn

Figure 18 shows the experimental values of these parameters for the observations in1982 at site I Interesting deviations from the time-averaged values (Fig 5) occur themost noteworthy being the unexpected high cyclonic frictional depths between day220-230 and 235-240 (Fig 18a) Variations in Ekman depth b+ for a particularfrequency (J need not necessarily be produced by changes in the effective Coriolisfrequency but may equally well result from time variations in the turbulent state of thesea Yet a simple increase of background turbulent eddy viscosity (eg due to the

1987]

5Maas amp van Haren Vertical current structure

90

315

s 1

I 0 240220 230 250

A~e

o

-90 220 230 240 250YEAR-DAY

Figure 18 Slow time evolution of friction depths 5plusmn (a) and phase differences between upperand lower current meters 60 plusmn (b) determined for the rotary current components at position 1for the semidiurnal frequency band The cyclonic component (+) is dashed the anticycloniccomponent (-) is solid Note that 5 was set zero if it was below the normalized depth of thelowest current meter

spring-neap tidal cycle) is not a sufficient explanation since this would correspond-ingly increase the anticyclonic depth 5_ Changes in the effective Coriolis frequencyhowever have opposite effects on 5+ and 5_ as follows from Figure 17 Sufficientlynegative values of the effective Coriolis frequency may bring about a (near) resonanceof the cyclonically rotating semidiurnal current component w+ and hence explain theobserved high values of 5+ This explanation is plausible since (near) resonances of thecyclonic and anticyclonic components 5plusmn seem to be complementary implying awandering of the effective Coriolis frequency in the order of plusmnf an unexpectedly largevalue (compare with oceanic values in warm-core rings which reach magnitudes of05 x f Joyce and Kennelly 1985) The observed phase differences 68plusmnpartly supportthis interpretation They are not in phase with the frictional depth variations but this isnot to be expected as we note from Figure 17 by observing how 5 and 68vary onchanges of v =fo plusmn a Note in Figure 18b how up to day 238 68+ follows 68_ almostexactly Recall that after day 240 the data become less reliable

If the observed slowtime behavior of5plusmn and 68is due to variations of 0(10-4 S-I) inthe effective Coriolis parameter this implies that we must expect strong current shearsat small scales (say changes of 10 cm S-1 over 0(1 km) distances)These scales arebelow our observational mooring separation distances and hence we cannot verify oursuggestion directly It would however be interesting to check the occurrence of suchsmall-scale relative vorticity which can possibly be ascribed to baroclinic eddiestrapped to the frontal zone whose presence is suggested in Van Aken et al (1987)


6 ConclusionsThe vertical structure of tidal currents can be explained physically in terms of

Ekman dynamics by considering force balances in frames rotating with the net angularvelocities (f plusmn (1)2 in which the rotary currents reduce to steady currents Amplitudeprofiles of the rotary current components in the semidiurnal frequency band show thepredicted separation in Ekman scales oplusmn experimentally Also phases for these rotarycurrent components turn in opposite (similar) directions with depth for superinertial(subinertial) motions again in accordance with theory The amount of agreementbetween observed and theoretical profiles depends on the signal-to-noise ratio of aspecific tidal component A fit of the theoretical curves to the observed vertical profilesfor the M2 frequency allows an experimental determination of an overall mean value ofthe vertical eddy viscosity (K (14 plusmn 01) x 10-3m2 s-I)and drag coefficient(Cd = (25 plusmn 02) x 10-3) of which the fonner is slightly less than values estimated inthe literature Near the bottom the largest deviations occur probably because of a localdecrease in eddy viscosity K due to a decreasing mixing length The values thusobtained are confirmed by their application to theoretical profiles at other tidal bandsFor the area under consideration and the time span of typically one month on whichthe observations are based the implication is that a constant K describes theturbulence state well

Vertical eddy viscosity variations were observed by comparing amplitude and phasevariations between two short periods (15 days) having a marked difference instratification The stratification influence is restricted to the anticyclonic componentand consists of a strong phase and amplitude jump at the bottom and interior of thepycnocline together with an amplitude decrease in the upper layer toward the surfaceThe former aspect is well described using a three layer model with a smaller eddyviscosity within the pycnocline modelling the local decrease of turbulence there Theamplitude decrease toward the surface is attributed to a small amount of stress fromthe air experienced by the moving water at the surface This also suggests a negligibleinfluence of the wind in the semidiurnal spectral band

Inertial motions were frictionally-modified first baroclinic mode probably gener-ated by the passage of storms The small phase differences between current meters atdifferent moorings but with comparable depths indicate a large wavelength Theinertial motions appear to experience some trapping in a passing front due to a localminimum in the effective Coriolis frequency ie the Coriolis frequency modified bylow-frequency relative vorticity

Variations in the effective Coriolis frequency are also suggested by the verticalstructure of inertial currents as well as by the experimentally determined slow timeevolution of frictional depth and phase difference of the two counterrotating semidiur-nal current components Events showing large values of the cyclonic friction depthimply the presence of strong small-scale vorticity of 0(10-4 S-I) associated with frontsand eddies

1987) Moos amp van Haren Vertical current structure 317

Acknowledgments This study was partly performed at the Institute of Meteorology andOceanography Utrecht the Netherlands The authors would like to express their gratitude toH M Van Aken of this institute for introducing them to the numerical aspects of tidal analysisto HW Riepma of the Royal Netherlands Meteorological Institute for kindly supplying themwith the current meter data records and to J T F Zimmerman of the Netherlands Institute forSea Research for his stimulating comments on an earlier draft The valuable comments of areferee which helped to improve our manuscript are appreciated The authors are supported bya grant from the Netherlands Organization for the Advancement of Pure Research (ZWO)

REFERENCESBowden K F 1983 Physical Oceanography of Coastal Waters Ellis Horwood Limited

Chicester 302 ppCsanady G T 1976 Mean circulation in shallow seas J Geophys Res 815389-5399-- 1982 Circulation in the Coastal Ocean D Reidel Publishing Company Dordrecht

BostonLondon 279 ppDronkers J J 1964 Tidal Computations in Rivers and Coastal Waters North Holland

Publishing Company Amsterdam 518 ppFang G and T Ichiye 1983 On the vertical structure of tidal currents in a homogeneous sea

Geophys J R Astr Soc 73 65-82Fjeldstad J E 1963 Internal waves of tidal origin Geor Publik 25 1-73Gargett A E and G Holloway 1984 Dissipation and diffusion by internal wave breaking J

Mar Res 4215-27Gill A E 1984 On the behaviour of internal waves in the wakes of storms J Phys Oceanogr

14566-581Joyce T M and M A Kennelly 1985 Upper-Ocean velocity structure of Gulf Stream

warm-core ring 82B J Geophys Res 90 (C5) 8839-8844Kunze E and T B Sanford 1984 Observations of near-inertial waves in a front J Phys

Oceanogr14 1129-1151Lamb Sir H 1975 Hydrodynamics Cambridge University Press Cambridge 738 ppLorentz H A 1926 Verslag Staatscommissie Zuiderzee 1916-1926 Algemene Landsdruk-

kerij s-Gravenhage 1-345Millot C and M Crepon 1981 Inertial oscillations on the continental shelf of the Gulf of

Lions-Observations and Theory J Phys Oceanogr 11 639-657Mooers C N K 1975 Several effects of a baroclinic current on the cross-stream propagation of

inertial-internal waves Geophys Fluid Dyn 6245-275Pedlosky J 1979 Geophysical Fluid Dynamics Springer Verlag New YorkHeidelberg

Berlin 624 ppPollard R T and R C Millard 1970 Comparison between observed and simulated

wind-generated inertial oscillations Deep-Sea Res 17 813-821Prandle D 1982 The vertical structure of tidal currents Geophys and Astrophys Fluid Dyn

2229-49Schott F 1971 Spatial structure of inertial-period motions in a two-layered sea based on

observations J Mar Res 29 85-102-- 1977 On the energetics of baroclinic tides in the North Atlantic Ann Geophys 33

41-62Soulsby R L 1983 The bottom boundary layer of shelf seas in Physical Oceanography of

Coastal and Shelf Seas B Johns ed Elsevier Scientific Publishing Company AmsterdamOxfordNew YorkTokyo 189-266


Sverdrup H U 1927 Dynamics oftides on the north Siberian Shelf Geof Publik 43-75Tee K-T 1979The structure of three-dimensional tide generating currents Part I oscillating

currents J Phys Oceanogr 9 930-944-- 1982The structure of three-dimensional tide-generating currents experimental verifica-

tion ofa theoretical model Est Coast Shelf Sci 14 27-48Van Aken H M 1984A one-dimensional mixed-layer model for stratified shelf seas with tide-

and wind-induced mixing Dt Hydrogr Z 373-27-- 1986The onset of seasonal stratification in shelf seas due to differential advection in the

presence of a salinity gradient Cont Shelf Res 5 475-485Van Aken H M G J F van Heijst and L R M Maas 1987Observations of fronts in the

North Sea J Mar Res (in press)Wimbush M and W Munk 1970 The benthic boundary layer in The Sea Vol 4 Wiley

731-758Wunsch C 1975 Internal tides Rev Geophys Space Phys 13 167-182

Received 30 July 1986 revised 16 December 1986


crt-t=O (f) +

Figure 1 Decomposition of a tidal current ellipse (frequency IT) specified by the four ellipseparameters magnitude of semi-major axis V eccentricity e - VIV (V semi-minor axis)inclination 11 and phase angle fjJ into two counterrotating currents of constant magnitudes Wand directions (J bullbull

Following Prandle (1982) to understand the vertical structure of these fourparameters we decompose the ellipsoidal motion into two counterrotating circularvelocity components with fixed amplitudes (the radii W) and phases (0) Figure 1 interms of which the ellipse parameters read

u= W+ + W_

e = (W+ - W_)(W+ + W_)

1 = (0_ + 0+)2

cent = (8_ - 0+)2

Sverdrup (1927) pointed out that in terms of these rotary current parameters thegoverning (linear) equations are solved in a straightforward way It is conceptuallyadvantageous however to stress that this solution procedure (Section 2) consists oftwo steps Firstly the circular velocity components which themselves are given on auniformly rotatingf-plane are transformed to two co-rotating frames having differentangular velocities v2 = (f plusmn u)2 where f denotes the local inertial frequencySecondly since in their respective co-rotating frames the velocity vectors reduce tosteady currents merely giving the amplitude and angle with respect to an originallychosen direction Ekman dynamics can be applied directly This explicit separationallows us to obtain a qualitative mental picture of the causal relations underlying thevertical structure of the ellipse parameters

Thus for super inertial frequencies (u gt f) the net rotation sense of the frameco-rotating with the anticyclonic current component is negative (clockwise) Hencedynamics apply as if we are on the Southern hemisphere and therefore the anticyclonicvelocity vector will rotate clockwise toward the bottom For semidiurnal frequencies atmoderate latitudes (u lt f) a separation of Ekman layer scales 0 = y2K If plusmn ui(Soulsby 1983) is predicted 0+ laquo 0_ Here K denotes the turbulent eddy viscosityNear the bottom therefore the cyclonic current component will be less reduced than








at ax 2 az2


at ay 2 az2

~ + 7 bull (f U dZ) = 0 (Ie)


az

au(Ie)-=0 at z = 1

az





u + iiiw ~ -- = W exp (i8 )- 2 - -

u - iiiw = -- = W exp (- i8 )+ 2 + +

(2)





IE d2W~1(1 - u)w + - jr= ---~ 2 ~ 2 dz2


dw+-- =swdz plusmn

dwplusmn = 0dz

-1(1 ) I r E d2w++(JW +-j~=---

+ 2 2 dz2

at z = 0

at z = I

(4)

(5)

1987]

where



297


with

(6)

dw-=swdz

dw-=0dz

at z = 0

at z ~ 1

(7a)

(7b)



where

w = ivp



a = (I + i)E~2

(8)

(9)


where

(10)

(II)

298

and


a_ = [(1 - i) (Hjo_)(1 + i) (Hjo_)

a+ = [(1 - i) (Hjo+)(1 + i) (Hjo+)

poundI gt 1

1gt poundIf

poundIfgt -1

-1 gt poundIf (12)








N56deg

51degI I I I I I I



AD 8+1- 54deg46N

1980 10km

0+ c+J 54deg41N


deg 0 0It It ltt


EDI

16km

1982H~

54deg30 N

GDK 54027 N





A 54deg46 3deg38 16 46 240-289 198041 240-275

B 54deg46 3deg47 14 46 240-3052841 240-275

C 54deg41 3deg47 13 46 240-27541 240-273

D 54deg41 3deg38 27 46 240-27341 240-274


18 215-23944 215-240












12 m

18 m

24 m

o cmS

--

N

n

30 m

37 m

44 m

-



5







o

05

ZlH




0

o w_ 2 Iilt if

J)

bull to __

9_ liltf







304

ZHbull0

-



[452

oo W 2 o

w_ 2 -04 8_ 04







ZlH0

0

bull

e

oo W 2 o w_ 2 -04 8 04 -04 8_ 04



245 250

TIME (DAYS)

230 235

10

125120

11

12

215

]0

50

II


I10 I

1--------

20

N

nt------t5 d y n c m2

215 235YEAR-DAY 255



8_ rid

[452

I

bull tI

II lll171D1

if2 -iiiw_

II 11111 7lll

W 2

306






1987]

ZlHLO


O

oo It 2 It_ 2 -0

Omiddot e- d







Figures 11 and 12


ZlH10

oo

w 2

if

8 till









N

2 cms n 2~gt----lt


0 f

7~

-

(14)







N

12m ~ 0 0 018m C)

G3crrvS-

24m -Q--Q -------27m

30m

37m Q

44m C Q G

J K L

12m ~ lt2gt ()

Jcmls---24m

27m

30m

37m

44m () -I J K L

[452


1987]

ZlH10

5

00


L

20 -1 Bd 90middot

311

1 Bo

















1987]

~ 0Ev


YEAR-DAY

12m

313

18m

10

o

10

o

10

o

24m

JOm

37m

44m





10

5

0-100 -50 o

F50 100

-40middot-100 -50 o

F50 100





1987]


90

315

s 1

I 0 240220 230 250

A~e

o

-90 220 230 240 250YEAR-DAY














































at ax 2 az2


at ay 2 az2

~ + 7 bull (f U dZ) = 0 (Ie)


az

au(Ie)-=0 at z = 1

az





u + iiiw ~ -- = W exp (i8 )- 2 - -

u - iiiw = -- = W exp (- i8 )+ 2 + +

(2)





IE d2W~1(1 - u)w + - jr= ---~ 2 ~ 2 dz2


dw+-- =swdz plusmn

dwplusmn = 0dz

-1(1 ) I r E d2w++(JW +-j~=---

+ 2 2 dz2

at z = 0

at z = I

(4)

(5)

1987]

where



297


with

(6)

dw-=swdz

dw-=0dz

at z = 0

at z ~ 1

(7a)

(7b)



where

w = ivp



a = (I + i)E~2

(8)

(9)


where

(10)

(II)

298

and


a_ = [(1 - i) (Hjo_)(1 + i) (Hjo_)

a+ = [(1 - i) (Hjo+)(1 + i) (Hjo+)

poundI gt 1

1gt poundIf

poundIfgt -1

-1 gt poundIf (12)








N56deg

51degI I I I I I I



AD 8+1- 54deg46N

1980 10km

0+ c+J 54deg41N


deg 0 0It It ltt


EDI

16km

1982H~

54deg30 N

GDK 54027 N





A 54deg46 3deg38 16 46 240-289 198041 240-275

B 54deg46 3deg47 14 46 240-3052841 240-275

C 54deg41 3deg47 13 46 240-27541 240-273

D 54deg41 3deg38 27 46 240-27341 240-274


18 215-23944 215-240












12 m

18 m

24 m

o cmS

--

N

n

30 m

37 m

44 m

-



5







o

05

ZlH




0

o w_ 2 Iilt if

J)

bull to __

9_ liltf







304

ZHbull0

-



[452

oo W 2 o

w_ 2 -04 8_ 04







ZlH0

0

bull

e

oo W 2 o w_ 2 -04 8 04 -04 8_ 04



245 250

TIME (DAYS)

230 235

10

125120

11

12

215

]0

50

II


I10 I

1--------

20

N

nt------t5 d y n c m2

215 235YEAR-DAY 255



8_ rid

[452

I

bull tI

II lll171D1

if2 -iiiw_

II 11111 7lll

W 2

306






1987]

ZlHLO


O

oo It 2 It_ 2 -0

Omiddot e- d







Figures 11 and 12


ZlH10

oo

w 2

if

8 till









N

2 cms n 2~gt----lt


0 f

7~

-

(14)







N

12m ~ 0 0 018m C)

G3crrvS-

24m -Q--Q -------27m

30m

37m Q

44m C Q G

J K L

12m ~ lt2gt ()

Jcmls---24m

27m

30m

37m

44m () -I J K L

[452


1987]

ZlH10

5

00


L

20 -1 Bd 90middot

311

1 Bo

















1987]

~ 0Ev


YEAR-DAY

12m

313

18m

10

o

10

o

10

o

24m

JOm

37m

44m





10

5

0-100 -50 o

F50 100

-40middot-100 -50 o

F50 100





1987]


90

315

s 1

I 0 240220 230 250

A~e

o

-90 220 230 240 250YEAR-DAY










































u + iiiw ~ -- = W exp (i8 )- 2 - -

u - iiiw = -- = W exp (- i8 )+ 2 + +

(2)





IE d2W~1(1 - u)w + - jr= ---~ 2 ~ 2 dz2


dw+-- =swdz plusmn

dwplusmn = 0dz

-1(1 ) I r E d2w++(JW +-j~=---

+ 2 2 dz2

at z = 0

at z = I

(4)

(5)

1987]

where



297


with

(6)

dw-=swdz

dw-=0dz

at z = 0

at z ~ 1

(7a)

(7b)



where

w = ivp



a = (I + i)E~2

(8)

(9)


where

(10)

(II)

298

and


a_ = [(1 - i) (Hjo_)(1 + i) (Hjo_)

a+ = [(1 - i) (Hjo+)(1 + i) (Hjo+)

poundI gt 1

1gt poundIf

poundIfgt -1

-1 gt poundIf (12)








N56deg

51degI I I I I I I



AD 8+1- 54deg46N

1980 10km

0+ c+J 54deg41N


deg 0 0It It ltt


EDI

16km

1982H~

54deg30 N

GDK 54027 N





A 54deg46 3deg38 16 46 240-289 198041 240-275

B 54deg46 3deg47 14 46 240-3052841 240-275

C 54deg41 3deg47 13 46 240-27541 240-273

D 54deg41 3deg38 27 46 240-27341 240-274


18 215-23944 215-240












12 m

18 m

24 m

o cmS

--

N

n

30 m

37 m

44 m

-



5







o

05

ZlH




0

o w_ 2 Iilt if

J)

bull to __

9_ liltf







304

ZHbull0

-



[452

oo W 2 o

w_ 2 -04 8_ 04







ZlH0

0

bull

e

oo W 2 o w_ 2 -04 8 04 -04 8_ 04



245 250

TIME (DAYS)

230 235

10

125120

11

12

215

]0

50

II


I10 I

1--------

20

N

nt------t5 d y n c m2

215 235YEAR-DAY 255



8_ rid

[452

I

bull tI

II lll171D1

if2 -iiiw_

II 11111 7lll

W 2

306






1987]

ZlHLO


O

oo It 2 It_ 2 -0

Omiddot e- d







Figures 11 and 12


ZlH10

oo

w 2

if

8 till









N

2 cms n 2~gt----lt


0 f

7~

-

(14)







N

12m ~ 0 0 018m C)

G3crrvS-

24m -Q--Q -------27m

30m

37m Q

44m C Q G

J K L

12m ~ lt2gt ()

Jcmls---24m

27m

30m

37m

44m () -I J K L

[452


1987]

ZlH10

5

00


L

20 -1 Bd 90middot

311

1 Bo

















1987]

~ 0Ev


YEAR-DAY

12m

313

18m

10

o

10

o

10

o

24m

JOm

37m

44m





10

5

0-100 -50 o

F50 100

-40middot-100 -50 o

F50 100





1987]


90

315

s 1

I 0 240220 230 250

A~e

o

-90 220 230 240 250YEAR-DAY







































1987]

where



297


with

(6)

dw-=swdz

dw-=0dz

at z = 0

at z ~ 1

(7a)

(7b)



where

w = ivp



a = (I + i)E~2

(8)

(9)


where

(10)

(II)

298

and


a_ = [(1 - i) (Hjo_)(1 + i) (Hjo_)

a+ = [(1 - i) (Hjo+)(1 + i) (Hjo+)

poundI gt 1

1gt poundIf

poundIfgt -1

-1 gt poundIf (12)








N56deg

51degI I I I I I I



AD 8+1- 54deg46N

1980 10km

0+ c+J 54deg41N


deg 0 0It It ltt


EDI

16km

1982H~

54deg30 N

GDK 54027 N





A 54deg46 3deg38 16 46 240-289 198041 240-275

B 54deg46 3deg47 14 46 240-3052841 240-275

C 54deg41 3deg47 13 46 240-27541 240-273

D 54deg41 3deg38 27 46 240-27341 240-274


18 215-23944 215-240












12 m

18 m

24 m

o cmS

--

N

n

30 m

37 m

44 m

-



5







o

05

ZlH




0

o w_ 2 Iilt if

J)

bull to __

9_ liltf







304

ZHbull0

-



[452

oo W 2 o

w_ 2 -04 8_ 04







ZlH0

0

bull

e

oo W 2 o w_ 2 -04 8 04 -04 8_ 04



245 250

TIME (DAYS)

230 235

10

125120

11

12

215

]0

50

II


I10 I

1--------

20

N

nt------t5 d y n c m2

215 235YEAR-DAY 255



8_ rid

[452

I

bull tI

II lll171D1

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II 11111 7lll

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306






1987]

ZlHLO


O

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Figures 11 and 12


ZlH10

oo

w 2

if

8 till









N

2 cms n 2~gt----lt


0 f

7~

-

(14)







N

12m ~ 0 0 018m C)

G3crrvS-

24m -Q--Q -------27m

30m

37m Q

44m C Q G

J K L

12m ~ lt2gt ()

Jcmls---24m

27m

30m

37m

44m () -I J K L

[452


1987]

ZlH10

5

00


L

20 -1 Bd 90middot

311

1 Bo

















1987]

~ 0Ev


YEAR-DAY

12m

313

18m

10

o

10

o

10

o

24m

JOm

37m

44m





10

5

0-100 -50 o

F50 100

-40middot-100 -50 o

F50 100





1987]


90

315

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I 0 240220 230 250

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298

and


a_ = [(1 - i) (Hjo_)(1 + i) (Hjo_)

a+ = [(1 - i) (Hjo+)(1 + i) (Hjo+)

poundI gt 1

1gt poundIf

poundIfgt -1

-1 gt poundIf (12)








N56deg

51degI I I I I I I



AD 8+1- 54deg46N

1980 10km

0+ c+J 54deg41N


deg 0 0It It ltt


EDI

16km

1982H~

54deg30 N

GDK 54027 N





A 54deg46 3deg38 16 46 240-289 198041 240-275

B 54deg46 3deg47 14 46 240-3052841 240-275

C 54deg41 3deg47 13 46 240-27541 240-273

D 54deg41 3deg38 27 46 240-27341 240-274


18 215-23944 215-240












12 m

18 m

24 m

o cmS

--

N

n

30 m

37 m

44 m

-



5







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05

ZlH




0

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304

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oo W 2 o

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ZlH0

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245 250

TIME (DAYS)

230 235

10

125120

11

12

215

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N

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215 235YEAR-DAY 255



8_ rid

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306






1987]

ZlHLO


O

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Omiddot e- d







Figures 11 and 12


ZlH10

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w 2

if

8 till









N

2 cms n 2~gt----lt


0 f

7~

-

(14)







N

12m ~ 0 0 018m C)

G3crrvS-

24m -Q--Q -------27m

30m

37m Q

44m C Q G

J K L

12m ~ lt2gt ()

Jcmls---24m

27m

30m

37m

44m () -I J K L

[452


1987]

ZlH10

5

00


L

20 -1 Bd 90middot

311

1 Bo

















1987]

~ 0Ev


YEAR-DAY

12m

313

18m

10

o

10

o

10

o

24m

JOm

37m

44m





10

5

0-100 -50 o

F50 100

-40middot-100 -50 o

F50 100





1987]


90

315

s 1

I 0 240220 230 250

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N56deg

51degI I I I I I I



AD 8+1- 54deg46N

1980 10km

0+ c+J 54deg41N


deg 0 0It It ltt


EDI

16km

1982H~

54deg30 N

GDK 54027 N





A 54deg46 3deg38 16 46 240-289 198041 240-275

B 54deg46 3deg47 14 46 240-3052841 240-275

C 54deg41 3deg47 13 46 240-27541 240-273

D 54deg41 3deg38 27 46 240-27341 240-274


18 215-23944 215-240












12 m

18 m

24 m

o cmS

--

N

n

30 m

37 m

44 m

-



5







o

05

ZlH




0

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304

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oo W 2 o

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ZlH0

0

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245 250

TIME (DAYS)

230 235

10

125120

11

12

215

]0

50

II


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1--------

20

N

nt------t5 d y n c m2

215 235YEAR-DAY 255



8_ rid

[452

I

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II lll171D1

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II 11111 7lll

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306






1987]

ZlHLO


O

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Figures 11 and 12


ZlH10

oo

w 2

if

8 till









N

2 cms n 2~gt----lt


0 f

7~

-

(14)







N

12m ~ 0 0 018m C)

G3crrvS-

24m -Q--Q -------27m

30m

37m Q

44m C Q G

J K L

12m ~ lt2gt ()

Jcmls---24m

27m

30m

37m

44m () -I J K L

[452


1987]

ZlH10

5

00


L

20 -1 Bd 90middot

311

1 Bo

















1987]

~ 0Ev


YEAR-DAY

12m

313

18m

10

o

10

o

10

o

24m

JOm

37m

44m





10

5

0-100 -50 o

F50 100

-40middot-100 -50 o

F50 100





1987]


90

315

s 1

I 0 240220 230 250

A~e

o

-90 220 230 240 250YEAR-DAY










































A 54deg46 3deg38 16 46 240-289 198041 240-275

B 54deg46 3deg47 14 46 240-3052841 240-275

C 54deg41 3deg47 13 46 240-27541 240-273

D 54deg41 3deg38 27 46 240-27341 240-274


18 215-23944 215-240












12 m

18 m

24 m

o cmS

--

N

n

30 m

37 m

44 m

-



5







o

05

ZlH




0

o w_ 2 Iilt if

J)

bull to __

9_ liltf







304

ZHbull0

-



[452

oo W 2 o

w_ 2 -04 8_ 04







ZlH0

0

bull

e

oo W 2 o w_ 2 -04 8 04 -04 8_ 04



245 250

TIME (DAYS)

230 235

10

125120

11

12

215

]0

50

II


I10 I

1--------

20

N

nt------t5 d y n c m2

215 235YEAR-DAY 255



8_ rid

[452

I

bull tI

II lll171D1

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II 11111 7lll

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306






1987]

ZlHLO


O

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Figures 11 and 12


ZlH10

oo

w 2

if

8 till









N

2 cms n 2~gt----lt


0 f

7~

-

(14)







N

12m ~ 0 0 018m C)

G3crrvS-

24m -Q--Q -------27m

30m

37m Q

44m C Q G

J K L

12m ~ lt2gt ()

Jcmls---24m

27m

30m

37m

44m () -I J K L

[452


1987]

ZlH10

5

00


L

20 -1 Bd 90middot

311

1 Bo

















1987]

~ 0Ev


YEAR-DAY

12m

313

18m

10

o

10

o

10

o

24m

JOm

37m

44m





10

5

0-100 -50 o

F50 100

-40middot-100 -50 o

F50 100





1987]


90

315

s 1

I 0 240220 230 250

A~e

o

-90 220 230 240 250YEAR-DAY








































12 m

18 m

24 m

o cmS

--

N

n

30 m

37 m

44 m

-



5







o

05

ZlH




0

o w_ 2 Iilt if

J)

bull to __

9_ liltf







304

ZHbull0

-



[452

oo W 2 o

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ZlH0

0

bull

e

oo W 2 o w_ 2 -04 8 04 -04 8_ 04



245 250

TIME (DAYS)

230 235

10

125120

11

12

215

]0

50

II


I10 I

1--------

20

N

nt------t5 d y n c m2

215 235YEAR-DAY 255



8_ rid

[452

I

bull tI

II lll171D1

if2 -iiiw_

II 11111 7lll

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306






1987]

ZlHLO


O

oo It 2 It_ 2 -0

Omiddot e- d







Figures 11 and 12


ZlH10

oo

w 2

if

8 till









N

2 cms n 2~gt----lt


0 f

7~

-

(14)







N

12m ~ 0 0 018m C)

G3crrvS-

24m -Q--Q -------27m

30m

37m Q

44m C Q G

J K L

12m ~ lt2gt ()

Jcmls---24m

27m

30m

37m

44m () -I J K L

[452


1987]

ZlH10

5

00


L

20 -1 Bd 90middot

311

1 Bo

















1987]

~ 0Ev


YEAR-DAY

12m

313

18m

10

o

10

o

10

o

24m

JOm

37m

44m





10

5

0-100 -50 o

F50 100

-40middot-100 -50 o

F50 100





1987]


90

315

s 1

I 0 240220 230 250

A~e

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-90 220 230 240 250YEAR-DAY








































5







o

05

ZlH




0

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304

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[452

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w_ 2 -04 8_ 04







ZlH0

0

bull

e

oo W 2 o w_ 2 -04 8 04 -04 8_ 04



245 250

TIME (DAYS)

230 235

10

125120

11

12

215

]0

50

II


I10 I

1--------

20

N

nt------t5 d y n c m2

215 235YEAR-DAY 255



8_ rid

[452

I

bull tI

II lll171D1

if2 -iiiw_

II 11111 7lll

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306






1987]

ZlHLO


O

oo It 2 It_ 2 -0

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Figures 11 and 12


ZlH10

oo

w 2

if

8 till









N

2 cms n 2~gt----lt


0 f

7~

-

(14)







N

12m ~ 0 0 018m C)

G3crrvS-

24m -Q--Q -------27m

30m

37m Q

44m C Q G

J K L

12m ~ lt2gt ()

Jcmls---24m

27m

30m

37m

44m () -I J K L

[452


1987]

ZlH10

5

00


L

20 -1 Bd 90middot

311

1 Bo

















1987]

~ 0Ev


YEAR-DAY

12m

313

18m

10

o

10

o

10

o

24m

JOm

37m

44m





10

5

0-100 -50 o

F50 100

-40middot-100 -50 o

F50 100





1987]


90

315

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I 0 240220 230 250

A~e

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-90 220 230 240 250YEAR-DAY








































0

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J)

bull to __

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304

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-



[452

oo W 2 o

w_ 2 -04 8_ 04







ZlH0

0

bull

e

oo W 2 o w_ 2 -04 8 04 -04 8_ 04



245 250

TIME (DAYS)

230 235

10

125120

11

12

215

]0

50

II


I10 I

1--------

20

N

nt------t5 d y n c m2

215 235YEAR-DAY 255



8_ rid

[452

I

bull tI

II lll171D1

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II 11111 7lll

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306






1987]

ZlHLO


O

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Figures 11 and 12


ZlH10

oo

w 2

if

8 till









N

2 cms n 2~gt----lt


0 f

7~

-

(14)







N

12m ~ 0 0 018m C)

G3crrvS-

24m -Q--Q -------27m

30m

37m Q

44m C Q G

J K L

12m ~ lt2gt ()

Jcmls---24m

27m

30m

37m

44m () -I J K L

[452


1987]

ZlH10

5

00


L

20 -1 Bd 90middot

311

1 Bo

















1987]

~ 0Ev


YEAR-DAY

12m

313

18m

10

o

10

o

10

o

24m

JOm

37m

44m





10

5

0-100 -50 o

F50 100

-40middot-100 -50 o

F50 100





1987]


90

315

s 1

I 0 240220 230 250

A~e

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-90 220 230 240 250YEAR-DAY







































304

ZHbull0

-



[452

oo W 2 o

w_ 2 -04 8_ 04







ZlH0

0

bull

e

oo W 2 o w_ 2 -04 8 04 -04 8_ 04



245 250

TIME (DAYS)

230 235

10

125120

11

12

215

]0

50

II


I10 I

1--------

20

N

nt------t5 d y n c m2

215 235YEAR-DAY 255



8_ rid

[452

I

bull tI

II lll171D1

if2 -iiiw_

II 11111 7lll

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306






1987]

ZlHLO


O

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Omiddot e- d







Figures 11 and 12


ZlH10

oo

w 2

if

8 till









N

2 cms n 2~gt----lt


0 f

7~

-

(14)







N

12m ~ 0 0 018m C)

G3crrvS-

24m -Q--Q -------27m

30m

37m Q

44m C Q G

J K L

12m ~ lt2gt ()

Jcmls---24m

27m

30m

37m

44m () -I J K L

[452


1987]

ZlH10

5

00


L

20 -1 Bd 90middot

311

1 Bo

















1987]

~ 0Ev


YEAR-DAY

12m

313

18m

10

o

10

o

10

o

24m

JOm

37m

44m





10

5

0-100 -50 o

F50 100

-40middot-100 -50 o

F50 100





1987]


90

315

s 1

I 0 240220 230 250

A~e

o

-90 220 230 240 250YEAR-DAY








































245 250

TIME (DAYS)

230 235

10

125120

11

12

215

]0

50

II


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1--------

20

N

nt------t5 d y n c m2

215 235YEAR-DAY 255



8_ rid

[452

I

bull tI

II lll171D1

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II 11111 7lll

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306






1987]

ZlHLO


O

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Figures 11 and 12


ZlH10

oo

w 2

if

8 till









N

2 cms n 2~gt----lt


0 f

7~

-

(14)







N

12m ~ 0 0 018m C)

G3crrvS-

24m -Q--Q -------27m

30m

37m Q

44m C Q G

J K L

12m ~ lt2gt ()

Jcmls---24m

27m

30m

37m

44m () -I J K L

[452


1987]

ZlH10

5

00


L

20 -1 Bd 90middot

311

1 Bo

















1987]

~ 0Ev


YEAR-DAY

12m

313

18m

10

o

10

o

10

o

24m

JOm

37m

44m





10

5

0-100 -50 o

F50 100

-40middot-100 -50 o

F50 100





1987]


90

315

s 1

I 0 240220 230 250

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-90 220 230 240 250YEAR-DAY








































8_ rid

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II lll171D1

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1987]

ZlHLO


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Figures 11 and 12


ZlH10

oo

w 2

if

8 till









N

2 cms n 2~gt----lt


0 f

7~

-

(14)







N

12m ~ 0 0 018m C)

G3crrvS-

24m -Q--Q -------27m

30m

37m Q

44m C Q G

J K L

12m ~ lt2gt ()

Jcmls---24m

27m

30m

37m

44m () -I J K L

[452


1987]

ZlH10

5

00


L

20 -1 Bd 90middot

311

1 Bo

















1987]

~ 0Ev


YEAR-DAY

12m

313

18m

10

o

10

o

10

o

24m

JOm

37m

44m





10

5

0-100 -50 o

F50 100

-40middot-100 -50 o

F50 100





1987]


90

315

s 1

I 0 240220 230 250

A~e

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-90 220 230 240 250YEAR-DAY







































1987]

ZlHLO


O

oo It 2 It_ 2 -0

Omiddot e- d







Figures 11 and 12


ZlH10

oo

w 2

if

8 till









N

2 cms n 2~gt----lt


0 f

7~

-

(14)







N

12m ~ 0 0 018m C)

G3crrvS-

24m -Q--Q -------27m

30m

37m Q

44m C Q G

J K L

12m ~ lt2gt ()

Jcmls---24m

27m

30m

37m

44m () -I J K L

[452


1987]

ZlH10

5

00


L

20 -1 Bd 90middot

311

1 Bo

















1987]

~ 0Ev


YEAR-DAY

12m

313

18m

10

o

10

o

10

o

24m

JOm

37m

44m





10

5

0-100 -50 o

F50 100

-40middot-100 -50 o

F50 100





1987]


90

315

s 1

I 0 240220 230 250

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-90 220 230 240 250YEAR-DAY








































ZlH10

oo

w 2

if

8 till









N

2 cms n 2~gt----lt


0 f

7~

-

(14)







N

12m ~ 0 0 018m C)

G3crrvS-

24m -Q--Q -------27m

30m

37m Q

44m C Q G

J K L

12m ~ lt2gt ()

Jcmls---24m

27m

30m

37m

44m () -I J K L

[452


1987]

ZlH10

5

00


L

20 -1 Bd 90middot

311

1 Bo

















1987]

~ 0Ev


YEAR-DAY

12m

313

18m

10

o

10

o

10

o

24m

JOm

37m

44m





10

5

0-100 -50 o

F50 100

-40middot-100 -50 o

F50 100





1987]


90

315

s 1

I 0 240220 230 250

A~e

o

-90 220 230 240 250YEAR-DAY








































N

2 cms n 2~gt----lt


0 f

7~

-

(14)







N

12m ~ 0 0 018m C)

G3crrvS-

24m -Q--Q -------27m

30m

37m Q

44m C Q G

J K L

12m ~ lt2gt ()

Jcmls---24m

27m

30m

37m

44m () -I J K L

[452


1987]

ZlH10

5

00


L

20 -1 Bd 90middot

311

1 Bo

















1987]

~ 0Ev


YEAR-DAY

12m

313

18m

10

o

10

o

10

o

24m

JOm

37m

44m





10

5

0-100 -50 o

F50 100

-40middot-100 -50 o

F50 100





1987]


90

315

s 1

I 0 240220 230 250

A~e

o

-90 220 230 240 250YEAR-DAY








































N

12m ~ 0 0 018m C)

G3crrvS-

24m -Q--Q -------27m

30m

37m Q

44m C Q G

J K L

12m ~ lt2gt ()

Jcmls---24m

27m

30m

37m

44m () -I J K L

[452


1987]

ZlH10

5

00


L

20 -1 Bd 90middot

311

1 Bo

















1987]

~ 0Ev


YEAR-DAY

12m

313

18m

10

o

10

o

10

o

24m

JOm

37m

44m





10

5

0-100 -50 o

F50 100

-40middot-100 -50 o

F50 100





1987]


90

315

s 1

I 0 240220 230 250

A~e

o

-90 220 230 240 250YEAR-DAY







































1987]

ZlH10

5

00


L

20 -1 Bd 90middot

311

1 Bo

















1987]

~ 0Ev


YEAR-DAY

12m

313

18m

10

o

10

o

10

o

24m

JOm

37m

44m





10

5

0-100 -50 o

F50 100

-40middot-100 -50 o

F50 100





1987]


90

315

s 1

I 0 240220 230 250

A~e

o

-90 220 230 240 250YEAR-DAY
















































1987]

~ 0Ev


YEAR-DAY

12m

313

18m

10

o

10

o

10

o

24m

JOm

37m

44m





10

5

0-100 -50 o

F50 100

-40middot-100 -50 o

F50 100





1987]


90

315

s 1

I 0 240220 230 250

A~e

o

-90 220 230 240 250YEAR-DAY








































10

5

0-100 -50 o

F50 100

-40middot-100 -50 o

F50 100





1987]


90

315

s 1

I 0 240220 230 250

A~e

o

-90 220 230 240 250YEAR-DAY







































1987]


90

315

s 1

I 0 240220 230 250

A~e

o

-90 220 230 240 250YEAR-DAY







































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